Our project

We decided to do central Colombia, basically because it is where the capital is.

We built a model for the number of confirmed cases using all the others covariates (plus some we created) and we estimated the predictive accuracy of our selected model.

We decided to consider as central Colombia the following departments/districts: Bogotà DC, Boyacá, Tolima, Cundinamarca, Meta, Quindío, Valle del Cauca, Risaralda, Celdas, Boyacá, Antioquia, Santander, Casanare.

Loading the dataset

This dataset is missing completely the department Valle del Cauca!

##   ID de caso Fecha de diagnóstico Ciudad de ubicación Departamento o Distrito
## 1          1           06/03/2020         Bogotá D.C.             Bogotá D.C.
## 3          3           09/03/2020            Medellín               Antioquia
## 4          4           11/03/2020            Medellín               Antioquia
## 5          5           11/03/2020            Medellín               Antioquia
## 6          6           11/03/2020              Itagüí               Antioquia
##     Atención Edad Sexo        Tipo País de procedencia
## 1 Recuperado   19    F   Importado              Italia
## 3 Recuperado   50    F   Importado              España
## 4 Recuperado   55    M Relacionado                 Nan
## 5 Recuperado   25    M Relacionado                 Nan
## 6 Recuperado   27    F Relacionado                 Nan

Description of variables

  • ID de caso: ID of the confirmed case.

  • Fecha de diagnóstico: Date in which the disease was diagnosed.

  • Ciudad de ubicación: City where the case was diagnosed.

  • Departamento o Distrito: Department or district where the city belongs to.

  • Atención: Situation of the patient: recovered, at home, at the hospital, at the ICU or deceased.

  • Edad: Age of the confirmed case.

  • Sexo: Sex of the confirmed case.

  • Tipo: How the person got infected: in Colombia, abroad or unknown.

  • País de procedencia: Country of origin if the person got infected abroad.

Map

Here we can see our selected cities. The color of the pins is related with the number of cases: if they are less than \(10\) the color is “green”, if they are less than \(100\) the color is “orange”, otherwise it is “red”.

Preprocessing

We had to clean the dataset:

  • We transformed the Fecha de diagnóstico variable into a Date type variable,

  • we fixed the variable Id de caso (since we removed some departments, so some lines, the numbers weren’t consecutive),

  • we created a variable Grupo de edad,

  • we cleaned the column País de procedencia (replaced cities with the country) and created the variable Continente de procedencia (as the first is too fragmented we thought to consider the continents).

##    ID de caso Fecha de diagnóstico Ciudad de ubicación Departamento o Distrito
## 1           1           2020-03-06         Bogotá D.C.             Bogotá D.C.
## 2           2           2020-03-09            Medellín               Antioquia
## 3           3           2020-03-11            Medellín               Antioquia
## 4           4           2020-03-11            Medellín               Antioquia
## 5           5           2020-03-11              Itagüí               Antioquia
## 6           6           2020-03-11         Bogotá D.C.             Bogotá D.C.
## 7           7           2020-03-11         Bogotá D.C.             Bogotá D.C.
## 8           8           2020-03-12         Bogotá D.C.             Bogotá D.C.
## 9           9           2020-03-12         Bogotá D.C.             Bogotá D.C.
## 10         10           2020-03-13       Villavicencio                    Meta
##      Atención Edad Sexo        Tipo País de procedencia Grupo de edad
## 1  Recuperado   19    F   Importado              Italia         19_30
## 2  Recuperado   50    F   Importado              España         46_60
## 3  Recuperado   55    M Relacionado                 Nan         46_60
## 4  Recuperado   25    M Relacionado                 Nan         19_30
## 5  Recuperado   27    F Relacionado                 Nan         19_30
## 6  Recuperado   22    F   Importado              España         19_30
## 7  Recuperado   28    F   Importado              España         19_30
## 8  Recuperado   36    F   Importado              España         31_45
## 9  Recuperado   42    F   Importado              España         31_45
## 10 Recuperado   30    F   Importado              España         19_30

New dataset I

##          Date Elapsed time New cases/day Cumulative cases
## 1  2020-03-06            0             1                1
## 2  2020-03-09            3             1                2
## 3  2020-03-11            5             5                7
## 4  2020-03-12            6             2                9
## 5  2020-03-13            7             3               12
## 6  2020-03-14            8            14               26
## 7  2020-03-15            9            13               39
## 8  2020-03-16           10             8               47
## 9  2020-03-17           11            13               60
## 10 2020-03-18           12             9               69

New dataset II

##          Date Elapsed time Department Department ID New cases/day
## 1  2020-03-09            3  Antioquia             1             1
## 2  2020-03-11            5  Antioquia             1             3
## 3  2020-03-14            8  Antioquia             1             3
## 4  2020-03-15            9  Antioquia             1             1
## 5  2020-03-19           13  Antioquia             1             3
## 6  2020-03-20           14  Antioquia             1            11
## 7  2020-03-21           15  Antioquia             1             3
## 8  2020-03-22           16  Antioquia             1             5
## 9  2020-03-23           17  Antioquia             1            22
## 10 2020-03-25           19  Antioquia             1             8
##    Cumulative cases/Department Mean age
## 1                            1 50.00000
## 2                            4 35.66667
## 3                            7 30.00000
## 4                            8 55.00000
## 5                           11 52.33333
## 6                           22 39.81818
## 7                           25 31.00000
## 8                           30 45.40000
## 9                           52 36.45455
## 10                          60 29.75000

Exploring the dataset

Scattered infos about pandemic in Colombia (https://en.wikipedia.org/wiki/COVID-19_pandemic_in_Colombia):

  • the quarantine started on the 20th of March, since our data are from 6th of March to 2nd of April, it is very likeliy that quarantine effects are not witnessed in our data.

  • on March the 26th there was a damage in the machine that prepared the samples for processing and subsequent diagnosis of COVID-19, which affected the speed at which results were being produced. This could explain the very low number of confirmed cases.

## List of 3
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##   ..$ arrow        : logi FALSE
##   ..$ inherit.blank: logi FALSE
##   ..- attr(*, "class")= chr [1:2] "element_line" "element"
##  $ panel.background: list()
##   ..- attr(*, "class")= chr [1:2] "element_blank" "element"
##  $ panel.grid      : list()
##   ..- attr(*, "class")= chr [1:2] "element_blank" "element"
##  - attr(*, "class")= chr [1:2] "theme" "gg"
##  - attr(*, "complete")= logi FALSE
##  - attr(*, "validate")= logi TRUE

The previous plot represents the daily incidence of the desease across all the departments we are taking into account.

Frequency of cases across Departments

The major number of cases are in the capital Bogotà.

Cumulative number of confirmed cases

Here the growth seems exponential (and this is consistent with the fact that we are studying the early stages of the outbreak).

Variables plots

Age-Sex plot

There isn’t much difference between the sexes among the different group of ages.

Tipo plot

I think that en estudio means that it is not clear while the case is imported or not, however it seems like there are more imported cases, we can count them:

## # A tibble: 3 x 3
##   Tipo        `Total number` Percentage
##   <chr>                <int> <chr>     
## 1 Relacionado           8570 12%       
## 2 Importado              687 1%        
## 3 En Estudio           60406 87%

The majority of the cases in the country are people that got infected inside Colombia. Then, people that contracted the disease abroad came mainly from Europe, followed by North America and Central America.

Correlation between the categorical variables

We used Cramer’s V to calculate the correlation between our categorical variables.

The frequentist approach

Train/test split

We splitted the data so to leave out the last three points for prediction, because we have few points and because in this models it has no sense to predict a week, because the situation changes really fast.

GLM

Poisson

Poisson with Elapsed time as predictor for Cumulative cases

## [1] "Estimated overdispersion 126.060786858437"
## [1] "RMSE: 1760.50922940986"
## [1] "AIC: 21915.0291925917"
## [1] "Null deviance:  1740689.09"  "Residual deviance: 20716.39"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.158730158730159"

Prediction interval for test set

Poisson with Elapsed time as predictor for New cases/day

## [1] "Estimated overdispersion 50.354334425333"
## [1] "RMSE: 58886.3428634314"
## [1] "AIC: 7291.77457395168"
## [1] "Null deviance:  64369.08"   "Residual deviance: 6442.53"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0"

Prediction interval for test set

Poisson with Elapsed time plus Elapsed time^2 as predictor for Cumulative cases

## [1] "Estimated overdispersion 75.2646213768741"
## [1] "RMSE: 6940.37335779742"
## [1] "AIC: 12062.7254007421"
## [1] "Null deviance:  1740689.09"  "Residual deviance: 10862.08"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0"

Prediction interval for test set

Poisson with Elapsed time plus Elapsed time^2 as predictor for New cases/day

## [1] "Estimated overdispersion 292693.098510483"
## [1] "RMSE: 739.356869793342"
## [1] "AIC: 7291.99073697742"
## [1] "Null deviance:  64369.08"   "Residual deviance: 6440.75"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0"

Prediction interval for test set

Poisson with Elapsed time plus Sex for Cumulative cases

## [1] "RMSE: 3892.80826581986"
## [1] "AIC: 20887.674847589"
## [1] "Null deviance:  1740689.09"  "Residual deviance: 19687.03"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.158730158730159"

Prediction interval for test set

Poisson with Elapsed time plus Sex as predictor for New cases/day

## [1] "Estimated overdispersion 314252.241298368"
## [1] "RMSE: 1810.41842007254"
## [1] "AIC: 4079.27639935841"
## [1] "Null deviance:  64369.08"   "Residual deviance: 3228.03"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0"

Prediction interval for test set

Poisson with Elapsed time plus Age group for Cumulative cases

## [1] "Estimated overdispersion 439618.33736315"
## [1] "RMSE: 3249.31716470375"
## [1] "AIC: 20650.0253613547"
## [1] "Null deviance:  1740689.09"  "Residual deviance: 19441.38"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.119047619047619"

Prediction interval for test set

Poisson with Elapsed time plus Age group as predictor for New cases/day

## [1] "Estimated overdispersion 315886.748769649"
## [1] "RMSE: 1948.8032473443"
## [1] "AIC: 3845.15073713936"
## [1] "Null deviance:  64369.08"   "Residual deviance: 2985.91"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0"

Prediction interval for test set

Poisson with Elapsed time plus Department for Cumulative cases

## [1] "Estimated overdispersion 489833.895411907"
## [1] "RMSE: 3328.87135693358"
## [1] "AIC: 19275.3910967978"
## [1] "Null deviance:  1740689.09"  "Residual deviance: 18054.75"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.0793650793650794"

Prediction interval for test set

Poisson with Elapsed time plus Department as predictor for New cases/day

## [1] "Estimated overdispersion 354075.219540599"
## [1] "RMSE: 1447.54763192187"
## [1] "AIC: 2989.46506265985"
## [1] "Null deviance:  64369.08"   "Residual deviance: 2118.22"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0"

Prediction interval for test set

Poisson with Elapsed time, Age group and Departments as predictors for Cumulative cases

## [1] "Estimated overdispersion 547354.078180419"
## [1] "RMSE: 8243.70282563776"
## [1] "AIC: 18085.2709276108"
## [1] "Null deviance:  1740689.09"  "Residual deviance: 16854.63"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.119047619047619"

Prediction interval for test set

Poisson with Elapsed time, Age group and Departments as predictors for New cases/day

## [1] "Estimated overdispersion 6719176.07903642"
## [1] "RMSE: 1225.26041309972"
## [1] "AIC: 2860.38819023372"
## [1] "Null deviance:  64369.08"   "Residual deviance: 1979.14"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.436507936507937"

Prediction interval for test set

Work in progress.. the plot came out very strange, if we don’t plan to include it in the presentation it is not worth spending time in it.. also because in the previous plot the interval is strangely visible!

Poisson with Elapsed time, Elapsed time^2, Age group and Departments as predictors for Cumulative cases

## [1] "Estimated overdispersion 1238590.22360691"
## [1] "RMSE: 8441.28307878747"
## [1] "AIC: 8781.84768119021"
## [1] "Null deviance:  1740689.09" "Residual deviance: 7549.2"

Predictive accuracy

## [1] "Frequency of coverage: 0.0416666666666667"

Prediction interval for test set

Same obs as before.. there is something strange in the function when the model overestimates the predictions

Poisson with Elapsed time, Elapsed time^2, Age group and Departments as predictors for New cases/day

## [1] "Estimated overdispersion 14212701.7733773"
## [1] "RMSE: 3072.65556331751"
## [1] "AIC: 2225.47391742673"
## [1] "Null deviance:  64369.08"   "Residual deviance: 1342.23"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.452380952380952"

Prediction interval for test set

Same as before.

Summary table - Poisson models for Cumulative cases

Model AIC RMSE
Cumulative cases ~ Elapsed time 21915.03 1760.51
Cumulative cases ~ Elapsed time + Elapsed time^2 12062.73 6940.37
Cumulative cases ~ Elapsed time + Sex 20887.67 3892.81
Cumulative cases ~ Elapsed time + Age group 20650.03 3249.32
Cumulative cases ~ Elapsed time + Department 19275.39 3328.87
Cumulative cases ~ Elapsed time + Age group + Department 18085.27 8243.7
Cumulative cases ~ Elapsed time + Elapsed time^2 + Age group + Department 8781.85 8441.28

Summary table - Poisson models for New cases/day

Model AIC RMSE
New cases/day ~ Elapsed time 7291.77 58886.34
New cases/day ~ Elapsed time + Elapsed time^2 7291.99 739.36
New cases/day ~ Elapsed time + Sex 4079.28 1810.42
New cases/day ~ Elapsed time + Age group 3845.15 1948.8
New cases/day ~ Elapsed time + Department 2989.47 1447.55
New cases/day ~ Elapsed time + Age group + Department 2860.39 1225.26
New cases/day ~ Elapsed time + Elapsed time^2 + Age group + Department 2225.47 3072.66

Autocorrelation to compare Poisson models

We generated 1000 samples from each of the four Poisson models and calculated the autocorrelation and compared against the autocorrelation of our original sample.

Autocorrelation for the models when the response variable is Cumulative Cases

Autocorrelation for the models when the response variable is New cases/day

ANOVA to compare the Poisson models

## Analysis of Deviance Table
## 
## Model 1: `Cumulative cases` ~ `Elapsed time`
## Model 2: `Cumulative cases` ~ `Elapsed time` + I(`Elapsed time`^2)
## Model 3: `Cumulative cases` ~ `Elapsed time` + Sexo_M
## Model 4: `Cumulative cases` ~ `Elapsed time` + `Grupo de edad_19_30` + 
##     `Grupo de edad_31_45` + `Grupo de edad_46_60` + `Grupo de edad_60_75` + 
##     `Grupo de edad_76+`
## Model 5: `Cumulative cases` ~ `Elapsed time` + `Departamento o Distrito_Bogotá D.C.` + 
##     `Departamento o Distrito_Boyacá` + `Departamento o Distrito_Caldas` + 
##     `Departamento o Distrito_Casanare` + `Departamento o Distrito_Cauca` + 
##     `Departamento o Distrito_Cundinamarca` + `Departamento o Distrito_Meta` + 
##     `Departamento o Distrito_Quindío` + `Departamento o Distrito_Risaralda` + 
##     `Departamento o Distrito_Santander` + `Departamento o Distrito_Tolima`
## Model 6: `Cumulative cases` ~ `Elapsed time` + `Grupo de edad_19_30` + 
##     `Grupo de edad_31_45` + `Grupo de edad_46_60` + `Grupo de edad_60_75` + 
##     `Grupo de edad_76+` + `Departamento o Distrito_Bogotá D.C.` + 
##     `Departamento o Distrito_Boyacá` + `Departamento o Distrito_Caldas` + 
##     `Departamento o Distrito_Casanare` + `Departamento o Distrito_Cauca` + 
##     `Departamento o Distrito_Cundinamarca` + `Departamento o Distrito_Meta` + 
##     `Departamento o Distrito_Quindío` + `Departamento o Distrito_Risaralda` + 
##     `Departamento o Distrito_Santander` + `Departamento o Distrito_Tolima`
## Model 7: `Cumulative cases` ~ `Elapsed time` + I(`Elapsed time`^2) + `Grupo de edad_19_30` + 
##     `Grupo de edad_31_45` + `Grupo de edad_46_60` + `Grupo de edad_60_75` + 
##     `Grupo de edad_76+` + `Departamento o Distrito_Bogotá D.C.` + 
##     `Departamento o Distrito_Boyacá` + `Departamento o Distrito_Caldas` + 
##     `Departamento o Distrito_Casanare` + `Departamento o Distrito_Cauca` + 
##     `Departamento o Distrito_Cundinamarca` + `Departamento o Distrito_Meta` + 
##     `Departamento o Distrito_Quindío` + `Departamento o Distrito_Risaralda` + 
##     `Departamento o Distrito_Santander` + `Departamento o Distrito_Tolima`
##   Resid. Df Resid. Dev Df Deviance  Pr(>Chi)    
## 1       118    20716.4                          
## 2       117    10862.1  1   9854.3 < 2.2e-16 ***
## 3       117    19687.0  0  -8824.9              
## 4       113    19441.4  4    245.6 < 2.2e-16 ***
## 5       107    18054.7  6   1386.6 < 2.2e-16 ***
## 6       102    16854.6  5   1200.1 < 2.2e-16 ***
## 7       101     7549.2  1   9305.4 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Quasi-Poisson

Quasi Poisson with Elapsed time as predictor for Cumulative cases

## [1] "Estimated overdispersion 3065.63883645847"
## [1] "RMSE: 1760.50922940986"
## [1] "Null deviance:  1740689.09"  "Residual deviance: 20716.39"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.849206349206349"

Quasi Poisson with Elapsed time as predictors for New cases/day

## [1] "Estimated overdispersion 7743403.90238763"
## [1] "RMSE: 753.966239397641"
## [1] "Null deviance:  64369.08"   "Residual deviance: 6442.53"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.183333333333333"

Prediction interval for test set

Quite horrible.. maybe unnecessary since the previous plot is good!

Quasi Poisson with Elapsed time and Elapsed time^2 as predictor for Cumulative cases

## [1] "Estimated overdispersion 10518.5904954718"
## [1] "RMSE: 6940.37335779742"
## [1] "Null deviance:  1740689.09"  "Residual deviance: 10862.08"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.849206349206349"

Prediction interval for the test set

Not necessary also in this case!

Quasi Poisson with Elapsed time and Elapsed time^2 as predictors for New cases/day

## [1] "Estimated overdispersion 7530190.35400279"
## [1] "RMSE: 739.356869793342"
## [1] "Null deviance:  64369.08"   "Residual deviance: 6440.75"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.25"

Quasi Poisson with Elapsed time and Sex as predictor for Cumulative cases

## [1] "Estimated overdispersion 3368.17928139048"
## [1] "RMSE: 3892.80826581986"
## [1] "Null deviance:  1740689.09"  "Residual deviance: 19687.03"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.857142857142857"

Prediction interval for test set

Not necessary!

### Quasi Poisson with Elapsed time and Sex as predictors for New cases/day
r quasipoisson3bis <- glm(`New cases/day` ~ `Elapsed time` + Sexo_M, data=data1[1:120, ], family=quasipoisson) 

## [1] "Estimated overdispersion 4627403.99712943"
## [1] "RMSE: 1810.41842007254"
## [1] "Null deviance:  64369.08"   "Residual deviance: 3228.03"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.225"

Quasi Poisson with Elapsed time and Age group as predictor for Cumulative cases

## [1] "Estimated overdispersion 3481.12259843348"
## [1] "RMSE: 3249.31716470375"
## [1] "Null deviance:  1740689.09"  "Residual deviance: 19441.38"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.865079365079365"

Prediction interval for test set

Almost like the previous one.. but maybe even here it is not necessary

Quasi Poisson with Elapsed time plus Age group as predictors for New cases/day

## [1] "Estimated overdispersion 4986548.55338898"
## [1] "RMSE: 1948.8032473443"
## [1] "Null deviance:  64369.08"   "Residual deviance: 2985.91"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.308333333333333"

Quasi Poisson with Elapsed time and Department as predictor for Cumulative cases

## [1] "Estimated overdispersion 3576.53352183996"
## [1] "RMSE: 3328.87135693358"
## [1] "Null deviance:  1740689.09"  "Residual deviance: 18054.75"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.873015873015873"

Quasi Poisson with Elapsed time plus Department as predictors for New cases/day

## [1] "Estimated overdispersion 6172812.15892504"
## [1] "RMSE: 1447.54763192187"
## [1] "Null deviance:  64369.08"   "Residual deviance: 2118.22"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.408333333333333"

Summary table - Quasi Poisson models for Cumulative cases

Model RMSE
Cumulative cases ~ Elapsed time 1760.51
Cumulative cases ~ Elapsed time + Elapsed time^2 6940.37
Cumulative cases ~ Elapsed time + Sex 3892.81
Cumulative cases ~ Elapsed time + Age group 3249.32
Cumulative cases ~ Elapsed time + Department 3328.87
Cumulative cases ~ Elapsed time + Department + Age group 8243.7
Cumulative cases ~ Elapsed time + Elapsed time^2 + Department + Age group 8441.28

Summary table - Quasi Poisson models for New cases/day

Model RMSE
New cases/day ~ Elapsed time 753.97

Negative Binomial

Negative Binomial with Elapsed time as predictor for Cumulative cases

## [1] "Estimated overdispersion 177.301055416919"
## [1] "RMSE: 1765.66109000166"
## [1] "AIC: 21911.1770461798"
## [1] "Null deviance:  1738722.15"  "Residual deviance: 20710.41"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.158730158730159"

Prediction interval for test set

Negative Binomial with Elapsed time as pedictor for New cases/day

## [1] "Estimated overdispersion 1.18509200751555"
## [1] "RMSE: 630.336144980397"
## [1] "AIC: 1445.58057844612"
## [1] "Null deviance:  916.37"    "Residual deviance: 136.75"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.936507936507937"

Prediction interval for test set

Negative Binomial with Elapsed time and Elapsed time^2 as predictors for Cumulative cases

## [1] "Estimated overdispersion 95.2440152631022"
## [1] "RMSE: 6946.45324124321"
## [1] "AIC: 12059.2678128481"
## [1] "Null deviance:  1739240.05"  "Residual deviance: 10856.53"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.126984126984127"

Prediction interval for test set

Negative Binomial with Elapsed time and Elapsed time^2 as pedictors for New cases/day

## [1] "Estimated overdispersion 1.1928471628252"
## [1] "RMSE: 1196.18328341049"
## [1] "AIC: 1435.02770364321"
## [1] "Null deviance:  1000.61"   "Residual deviance: 134.83"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.920634920634921"

Prediction interval for test set

Negative Binomial with Elapsed time and Sex as predictors for Cumulative cases

## [1] "Estimated overdispersion 171.547294318382"
## [1] "RMSE: 3898.50452431487"
## [1] "AIC: 20882.9362435937"
## [1] "Null deviance:  1738961.22"  "Residual deviance: 19680.18"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.158730158730159"

Prediction interval for test set

Negative Binomial with Elapsed time and Sex as pedictors for New cases/day

## [1] "Estimated overdispersion 1.22488855750542"
## [1] "RMSE: 1182.5740737675"
## [1] "AIC: 1441.87950405362"
## [1] "Null deviance:  979.37"    "Residual deviance: 139.09"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.928571428571429"

Prediction interval for test set

Negative Binomial with Elapsed time plus Age group as predictors for Cumulative cases

## [1] "Estimated overdispersion 178.172494675829"
## [1] "RMSE: 3253.44941446475"
## [1] "AIC: 20645.2695885301"
## [1] "Null deviance:  1738975.56"  "Residual deviance: 19434.52"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.126984126984127"

Prediction interval for test set

Negative Binomial with Elapsed time and Age group as pedictors for New cases/day

## [1] "Estimated overdispersion 1.29900880793691"
## [1] "RMSE: 1943.62684829872"
## [1] "AIC: 1444.376892835"
## [1] "Null deviance:  1027.67"   "Residual deviance: 139.39"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.928571428571429"

Prediction interval for test set

Negative Binomial with Elapsed time plus Department as predictors for Cumulative cases

## [1] "Estimated overdispersion 181.630453104304"
## [1] "RMSE: 3335.37126363787"
## [1] "AIC: 19270.2478115559"
## [1] "Null deviance:  1739087.22" "Residual deviance: 18047.5"

Predictive accuracy

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.0793650793650794"

Prediction interval for test set

Negative Binomial with Elapsed time and Departments as pedictors for New cases/day

## [1] "Estimated overdispersion 1.36033553203781"
## [1] "RMSE: 2185.73149313813"
## [1] "AIC: 1431.44459445254"
## [1] "Null deviance:  1334.76"   "Residual deviance: 145.73"

Predictive accuracy of the NB model for New cases/day

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.936507936507937"

Prediction interval for test set

Negative Binomial with Elapsed time, Age group and Departments as pedictors

## [1] "Estimated overdispersion 185.625787049954"
## [1] "RMSE: 8253.89918445986"
## [1] "AIC: 18080.4893928766"
## [1] "Null deviance:  1739076.35"  "Residual deviance: 16847.74"

Predictive accuracy of the Negative Binomial model

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.119047619047619"

Prediction interval for test set

Negative Binomial with Elapsed time, Age group and Departments as pedictors for New cases/day

## [1] "Estimated overdispersion 1.60944976076295"
## [1] "RMSE: 1379.56664915407"
## [1] "AIC: 1435.14403991019"
## [1] "Null deviance:  1416.29"   "Residual deviance: 146.46"

Predictive accuracy of the NB model for New cases/day

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.936507936507937"

Prediction interval for test set

One word.. horrible!

Negative Binomial with Elapsed time, Elapsed time^2, Age group and Departments as pedictors

## [1] "Estimated overdispersion 92.8003925284947"
## [1] "RMSE: 8445.61048902606"
## [1] "AIC: 8780.25417160988"
## [1] "Null deviance:  1739084.62" "Residual deviance: 7545.51"

Predictive accuracy of the Negative Binomial model

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.214285714285714"

Prediction interval for test set

Negative Binomial with Elapsed time, Elapsed time^2, Age group and Departments as pedictors for New cases/day

## [1] "Estimated overdispersion 1.6556918183569"
## [1] "RMSE: 9304.269996047"
## [1] "AIC: 1376.63703081839"
## [1] "Null deviance:  2432.64"   "Residual deviance: 149.28"

Predictive accuracy of the NB model for New cases/day

Predicting with a \(95\%\) confidence interval

## [1] "Frequency of coverage: 0.896825396825397"

Prediction interval for test set

Summary table - Negative Binomial models for Cumulative cases

Model AIC RMSE
Cumulative cases ~ Elapsed time 21911.18 1765.66
Cumulative cases ~ Elapsed time + Elapsed time^2 12059.27 6946.45
Cumulative cases ~ Elapsed time + Sex 20882.94 3898.5
Cumulative cases ~ Elapsed time + Age group 20645.27 3253.45
Cumulative cases ~ Elapsed time + Department 19270.25 3335.37
Cumulative cases ~ Elapsed time + Age group + Department 18080.49 8253.9
Cumulative cases ~ Elapsed time + Elapsed time^2 + Age group + Department 8780.25 8445.61

Summary table - Negative Binomial models for New cases/day

Model AIC RMSE
New cases/day ~ Elapsed time + Age group + Department 1445.58 630.34
New cases/day ~ Elapsed time + Elapsed time^2 1435.03 1196.18
New cases/day ~ Elapsed time + Sex 1441.88 1182.57
New cases/day ~ Elapsed time + Age group 1444.38 1943.63
New cases/day ~ Elapsed time + Department 1431.44 2185.73
New cases/day ~ Elapsed time + Age group + Department 1435.14 1379.57
New cases/day ~ Elapsed time + Elapsed time^2 + Age group + Department 1376.64 9304.27

Autocorrelation to compare Negative Binomial models

We generated 1000 samples from each of the four Negative Binomial models and calculated the autocorrelation and compared against the autocorrelation of our original sample.

Autocorrelation for the models when the response variable is Cumulative Cases

Autocorrelation for the model when the response variable is New cases/day

Applying ANOVA to compare the negative binomial models

## Likelihood ratio tests of Negative Binomial Models
## 
## Response: Cumulative cases
##                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Model
## 1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       Elapsed time
## 2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               `Elapsed time` + I(`Elapsed time`^2)
## 3                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            `Elapsed time` + Sexo_M
## 4                                                                                                                                                                                                                                                                                                                                                                                                                                               `Elapsed time` + `Grupo de edad_19_30` + `Grupo de edad_31_45` + `Grupo de edad_46_60` + `Grupo de edad_60_75` + `Grupo de edad_76+`
## 5                                                                                                                                                   `Elapsed time` + `Departamento o Distrito_Bogotá D.C.` + `Departamento o Distrito_Boyacá` + `Departamento o Distrito_Caldas` + `Departamento o Distrito_Casanare` + `Departamento o Distrito_Cauca` + `Departamento o Distrito_Cundinamarca` + `Departamento o Distrito_Meta` + `Departamento o Distrito_Quindío` + `Departamento o Distrito_Risaralda` + `Departamento o Distrito_Santander` + `Departamento o Distrito_Tolima`
## 6                       `Elapsed time` + `Grupo de edad_19_30` + `Grupo de edad_31_45` + `Grupo de edad_46_60` + `Grupo de edad_60_75` + `Grupo de edad_76+` + `Departamento o Distrito_Bogotá D.C.` + `Departamento o Distrito_Boyacá` + `Departamento o Distrito_Caldas` + `Departamento o Distrito_Casanare` + `Departamento o Distrito_Cauca` + `Departamento o Distrito_Cundinamarca` + `Departamento o Distrito_Meta` + `Departamento o Distrito_Quindío` + `Departamento o Distrito_Risaralda` + `Departamento o Distrito_Santander` + \n    `Departamento o Distrito_Tolima`
## 7 `Elapsed time` + I(`Elapsed time`^2) + `Grupo de edad_19_30` + `Grupo de edad_31_45` + `Grupo de edad_46_60` + `Grupo de edad_60_75` + `Grupo de edad_76+` + `Departamento o Distrito_Bogotá D.C.` + `Departamento o Distrito_Boyacá` + `Departamento o Distrito_Caldas` + `Departamento o Distrito_Casanare` + `Departamento o Distrito_Cauca` + `Departamento o Distrito_Cundinamarca` + `Departamento o Distrito_Meta` + `Departamento o Distrito_Quindío` + `Departamento o Distrito_Risaralda` + `Departamento o Distrito_Santander` + \n    `Departamento o Distrito_Tolima`
##      theta Resid. df    2 x log-lik.   Test    df   LR stat. Pr(Chi)
## 1 11252780       118      -21905.177                                
## 2 15281003       117      -12051.268 1 vs 2     1  9853.9092       0
## 3 12812156       117      -20874.936 2 vs 3     0 -8823.6684       1
## 4 12919543       113      -20629.270 3 vs 4     4   245.6667       0
## 5 13821345       107      -19242.248 4 vs 5     6  1387.0218       0
## 6 13728063       102      -18042.489 5 vs 6     5  1199.7584       0
## 7 13798917       101       -8740.254 6 vs 7     1  9302.2352       0

GAM

Gam with Elapsed time as covariate for Cumulative cases

## [1] "RMSE: 1282.65627040671"
## [1] "AIC: 1587.11794466718"

Prediction interval for test set

Intervals are very very very thin..

Gam with Elapsed time as covariate for New cases/day

## [1] "RMSE: 1121.41502755924"
## [1] "AIC: 6361.02810119354"

Prediction interval for test set

Gam with Elapsed time and Elapsed time^2 as covariate for Cumulative cases

## [1] "RMSE: 6047.00503603486"
## [1] "AIC: 1517.87827688164"

Prediction interval for test set

Gam with Elapsed time and Elapsed time^2 as covariate for New cases/day

## [1] "RMSE: 756.744620683535"
## [1] "AIC: 6341.56819625084"

Prediction interval for test set

Gam with Elapsed time and Sex as covariate for Cumulative cases

## [1] "RMSE: 2301.65489717491"
## [1] "AIC: 1525.13432986912"

Prediction interval for test set

Gam with Elapsed time and Sex as covariate for New cases/day

## [1] "RMSE: 1301.34999278548"
## [1] "AIC: 1239.36316647314"

Prediction interval for test set

Gam with Elapsed time and Age group as covariate for Cumulative cases

## [1] "RMSE: 911.823954532677"
## [1] "AIC: 1543.48204372495"

Prediction interval for test set

Prediction interval for test set

Prediction interval for test set

Prediction interval for test set

Prediction interval for test set

Prediction interval for test set

Prediction interval for test set

Prediction interval for test set

Summary table - GAM models for Cumulative cases

Model AIC RMSE
Cumulative cases ~ Elapsed time 1587.12 1282.66
Cumulative cases ~ Elapsed time + Elapsed time^2 1517.88 6047.01
Cumulative cases ~ Elapsed time + Sex 1525.13 2301.65
Cumulative cases ~ Elapsed time + Age group 1543.48 911.82
Cumulative cases ~ Elapsed time + Department 1517.1 704.14
Cumulative cases ~ Elapsed time + Age group + Department 1486.71 1038.67
Cumulative cases ~ Elapsed time + Elapsed time^2 + Age group + Department 1337.17 4599.76

Summary table - GAM models for New cases/day

Model AIC RMSE
New cases/day ~ Elapsed time 6361.03 1121.42
New cases/day ~ Elapsed time + Elapsed time^2 6341.57 756.74
New cases/day ~ Elapsed time + Sex 1239.36 1301.35
New cases/day ~ Elapsed time + Age group 1094.53 412.27
New cases/day ~ Elapsed time + Department 1095.72 398.09
New cases/day ~ Elapsed time + Age group + Department 1078.03 868.4
New cases/day ~ Elapsed time + Elapsed time^2 + Age group + Department 1761.98 1082.02

The Bayesian approach

Poisson regression for response variable Cumulative cases/Department

As a first attempt, we fit a simple Poisson regression: \[ ln(\lambda_i) = \alpha + \beta\cdot elapsed\_time_i \\ y_i \sim \mathcal{Poisson}(\lambda_i) \\ \alpha \sim \mathcal{N}(0,1) \\ \beta \sim \mathcal{N}(0.25,1) \] with \(i = 1,\dots,134\), being \(134\) the number of rows of our dataset, and \(y_i\) represents the number of cases.

For what concerns the stan program, we used the function poisson_log_rng to describe the distribution of \(y_i\), namely the number of cases each day and the function poisson_log_lpmf to specify the likelihood.

Posterior predictive check - Poisson Cumulative cases/Department

poisson_cumulative

The fit is not satisfactory, it is probably due to overdispersion, we can check the residuals to confirm this hypothesis.

poisson_cumulative_ecdf

Residual check - Poisson Cumulative cases/Department

poisson_cumulative_res

The variance of the residuals increases as the predicted value increase. The standardized residuals should have mean 0 and standard deviation 1 (hence the lines at \(+2\) and \(-2\) indicates approximate \(95\%\) error bounds).

The plot of the standardized residuals indicates a large amount of overdispersion.

Intervals - Poisson Cumulative cases/Department

poisson_cumulative_intervals

Accuracy - Poisson Cumulative cases/Department

poisson_cumulative_acc

Negative Binomial model for Cumulative cases /Department

We try to improve the previous model using the Negative Binomial model:

\[ ln(\lambda_i) = \alpha + \beta\cdot elapsed\_time_i \\ y_i \sim \mathcal{Negative Binomial}(\lambda_i, \phi) \\ \alpha \sim \mathcal{N}(0,1) \\ \beta \sim \mathcal{N}(0.25,1) \]

Where the parameter \(\phi\) is called precision and it is such that:

\[ E[y_i] = \lambda_i \\ Var[y_i] = \lambda_i + \frac{\lambda_i^2}{\phi} \]

again \(i=1,\dots,134\). As \(\phi \rightarrow \infty\) the negative binomial approaches the Poisson distribution.

The stan function that we use here are neg_binomial_2_log_rng to specify the distribution of \(y_i\) and the function neg_binomial_2_log_lpmf for the likelihood.

Posterior predictive check - NB Cumulative cases /Department

NB_cumulative

NB_cumulative_ecdf

Residual check - NB Cumulative cases /Department

NB_cumulative_res

The situation is better now, but still we have too many residuals outside the \(95\%\) interval.

Intervals - NB Cumulative cases /Department

NB_cumulative_intervals

Accuracy across departments - NB Cumulative cases /Department

NB_cumulative_acc

We should take into account the differences across departments.

Negative Binomial model for New cases/day

Posterior predictive check - NB New cases/day

NB_cases

NB_cases_ecdf

Residual check - NB New cases/day

NB_cases_res

The situation is better now, but still we have too many residuals outside the \(95\%\) interval.

Intervals - NB New cases/day

NB_cases_intervals

Accuracy across departments - NB New cases/day

NB_cases_acc

Multilevel Negative Binomial regression for Cumulative/Department

We try to fit the following model, which also includes Age as covariat:

\[ ln(\lambda_i) = \alpha + \beta_{time}\cdot elapsed\_time_i + \beta_{age}\cdot age \\ y_i \sim \mathcal{Negative Binomial}(\lambda_i, \phi) \\ \alpha \sim \mathcal{N}(0,1) \\ \beta_{time} \sim \mathcal{N}(0.5,1) \\ \beta_{age} \sim \mathcal{N}(0,1) \]

Posterior predictive check - multi NB Cumulative/Department

NB2_cumulative

NB2_cumulative_ecdf

Residual check - multi NB Cumulative/Department

NB2_cumulative_res

Intervals - - multi NB Cumulative/Department

NB2_cumulative_intervals

Accuracy across departments - multi NB Cumulative/Department

NB2_cumulative_acc

Hierarchical model for Cumulative/Department

In order to improve the fit, we fit a model with department-specific intercept term.

So the varying intercept model that we take into account is now:

\[ ln(\lambda_{i,d}) = \alpha_d + + \beta_{time}\cdot elapsed\_time_i + \beta_{age}\cdot age_i\\ \alpha_d \sim \mathcal{N}(\mu + \beta_{pop}\cdot pop_d + \beta_{sur}\cdot surface_d + \beta_{dens} \cdot density_d, \sigma_{\alpha})\\ y_i \sim \mathcal{Negative Binomial}(\lambda_{i,d}, \phi) \]

The priors used for the above model are the following:

\[ \beta_{time} \sim \mathcal{N}(0.5,1) \\ \beta_{age} \sim \mathcal{N}(0,1) \\ \psi \sim \mathcal{N}(0,1) \]

being \(\psi = [\beta_{pop}, \beta_{sur}, \beta_{dens}]\).

New dataset

We added the following covariats into the dataset:

  • People: millions of inhabitants for each region;

    • Surface: \(km^3\), extent of each region;

    • Density: \(\frac{people}{km^2}\), density of the population in each region.

The model is:

Posterior predictive check - hierarchical NB Cumulative/Department

hier_cumulative

hier_cumulative_ecdf

Residual check - hierarchical NB Cumulative/Department

hier_cumulative_res

Very few points are now outside the \(95\%\) confidence interval.

Intervals - hierarchical NB Cumulative/Department

hier_cumulative_intervals

Accuracy across departments - hierarchical NB Cumulative/Department

hier_cumulative_acc

We can clearly see that the accuracy across the departments has significantly increased with respect to the previous models.

LOOIC

The Leave-One-Out cross validation is a method for estimating pointwise out-of-sample prediction accuracy from a fitted Bayesian model using the log-likelihood evaluated at the posterior simulation of the parameters values.

Plot the looic to compare models:

looic